Awakening is currently the best starting point for new players. It makes the occasional reference to older games, but is more or less a stand-alone game. It has a casual mode so you can learn the game without fear of losing characters, you have enemy monsters that appear on the world map, giving you an infinite amount of EXP, and there's an mechanic called Pair Up that is, to put it plainly, overpowered. Unlike most games, there's also some post-game content in the form of DLC. (Which has an exponential increase in difficulty and writing quality.)

After Awakening, I recommend Sacred Stones, because it's the only other game in the series where you can go back and fight monsters on previous levels, once again giving you infinite EXP. It also prepares you for the rest of the series, in that it doesn't have a Casual Mode or a Pair Up system, you can't reclass characters, but there is a choice when promoting. There's also a limit to how many Supports you can have per character per playthrough. Instead of the Pair Up system there is a Rescue system, which allows a unit to grab another unit and pull them to safety. While a unit is carrying another, they suffer a 50% loss to Speed and Skill, so this is pretty much only for emergencies.

After Sacred Stones, I'd suggest FE7 (Just called Fire Emblem outside of Japan), because it shares almost every mechanic that Sacred Stones has and is a little more difficult. There isn't a branching class system, which is the case for the rest of the series.

After FE7, try Shadow Dragon. It's just a remake of the first game, and has a reclassing system.

After Shadow Dragon, try Path of Radiance. It has a skill system, but not like Awakening's, since there's no reclassing.

After Path of Radiance, try Radiant Dawn. It's Literally Path of Radiance 2, and it has a third tier of classes.

I can't say anything about the other games, because I haven't played them. You may have to emulate a lot of these games if you don't want to spend a good amount of money.

Also, FE:If is coming out eventually, and it may take Awakening's place as the best starting point for new players. I doubt it will though, since one side is supposed to be like a more traditional Fire Emblem game, while the other is meant to be closer to Awakening with world map monsters and infinite EXP.